Axiom ax-mulass 8135

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8093. Proofs should normally use mulass 8163 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 8030	. . . 4 class ℂ
3	1, 2	wcel 2202	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2202	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2202	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1004	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 8037	. . . . 5 class ·
10	1, 4, 9	co 6018	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6018	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6018	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6018	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1397	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8163